Exact Synthesis of Broadband Three-Line Baluns Hong-Ming Lee, Member, IEEE, and Chih-Ming Tsai, Member, IEEE

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1 140 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 Exact Synthesis of Broadband Three-Line Baluns Hong-Ming Lee, Member, IEEE, and Chih-Ming Tsai, Member, IEEE Abstract Three types of three-line baluns are studied in this paper. The equivalent circuits of these baluns are derived using the model of coupled transmission lines. The balun designs with Chebyshev response can then be exactly synthesized. A complete design procedure of the baluns with second-, third-, and fourth-order insertion loss functions is presented. Balun designs with wide bandwidths of more than 2 to 1 are achievable by this method. Some guidelines for practical realization are also suggested. A third-order balun design with a fractional bandwidth of 80% is given to demonstrate the proposed procedure. Index Terms Baluns, circuit synthesis, distributed parameter filters, impedance matching. I. INTRODUCTION I N MICROWAVE circuits such as balanced mixers, push pull amplifiers, and antennas, baluns are needed for the transformation between the balanced and unbalanced ports. Among the various kinds of balun, Marchand balun [1] is rather popular due to its outstanding amplitude and phase balance. The synthesis method of Marchand balun with ideal Chebyshev response had been presented in [2] and [3]. Planar realizations of Marchand balun are now widely used in microwave integrated circuits and microwave monolithic integrated circuits [4]. They are generally composed of two sections of two-coupled lines. The studies on their size reduction had also been reported in several studies (see [5] and [6]). On the other hand, a different class of planar baluns, which uses only one section of three-coupled lines, as shown in Fig. 1, promises a more compact realization. In this figure, port 1 is the unbalanced port. Port 2 and port 3 are the balanced ports. and are the additional input and output matching networks. The balun in Fig. 1(a), which is called a type-i balun in this paper, was studied in [7] and [8]. Its coaxial realization can be found as a type 4d balun in [9]. Besides the connection and terminations at the ends, this three-line balun was formed by direct combination of two identical two-coupled-line sections. Therefore, these three-coupled lines are symmetrical. However, this restriction is found unnecessary. The second and third kinds of three-line baluns are shown in Fig. 1(b) and (c), which are called type-ii and type-iii baluns in this paper, respectively. Coaxial baluns of these two types can be found as type 3 and type 4c baluns in [9]. A planar circuit implementation of type-iii balun was studied in [10]. Nevertheless, the equivalent circuits of these Manuscript received January 06, 2008; revised September 16, First published December 12, 2008; current version published January 08, This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC E The authors are with the Institute of Computer and Communication Engineering, Department of Electrical Engineering, National Cheng Kung University, Tainan 70101, Taiwan ( tsaic@mail.ncku.edu.tw). Digital Object Identifier /TMTT Fig. 1. Configurations of the three types of three-line baluns. (a) Type-I balun from [7]. (b) Type-II balun and (c) type-iii balun from [9] and [10]. two types of baluns are not fully addressed. Furthermore, the exact synthesis procedure for these three types of baluns is not yet available. In this paper, three-line baluns are studied using the network model of generalized TEM coupled lines. It is found that the coupled lines in type-i and type-ii baluns are not required to be symmetrical. Therefore, more configurations, especially those using multilayer structures, are possible. A synthesis procedure is then developed based on the derived equivalent circuit. With the additional input and output transmission-line sections, it is found that second-, third-, and fourth-order of Chebyshev responses can be realized. These baluns can be designed to have wide bandwidths of more than 2 to 1. Some design guidelines for practical realization are also presented. A design example of a third-order balun with a fractional bandwidth of 80% is given at the end of this paper to demonstrate the proposed synthesis method. II. EQUIVALENT CIRCUITS OF THREE-LINE BALUNS To derive the equivalent circuit, the network model of coupled transmission lines in [11] is used. The equivalent circuit of type-i three-line balun with its corresponding terminations is shown in Fig. 2. The inductors and thick lines represent short /$ IEEE

2 LEE AND TSAI: EXACT SYNTHESIS OF BROADBAND THREE-LINE BALUNS 141 Fig. 2. Representation of type-i balun using the network model of coupled transmission lines in [11]. ports need to be identical and the two transformers should be the same, except for a 180 phase difference. Since the three short-circuited stubs in Fig. 4(b) can be combined through the transformers to form a single short-circuited stub connected to port 1, they will not affect the condition of balanced outputs. Therefore, the sufficient conditions for type-i balun to have balanced outputs are (4) (5) The above equations imply that the capacitance between line 1 (line 3) and ground should be equal to that between line 2 (line 1) and line 3 (line 2), i.e., (6) (7) Fig. 3. Representation of the line capacitances per unit length with the cross section of three-coupled lines. circuited stubs and unit elements, respectively. The characteristic admittance is given by where is the propagation velocity of TEM waves in the medium, and is the element in the per-unit-length capacitance matrix defined by [12] where and are the charge per unit length and potential on the corresponding line, respectively. The diagonal terms in (2) are where is the line capacitance per unit length between line and ground, as depicted in Fig. 3. The circuit in Fig. 2 is simplified to that shown in Fig. 4(a). The outside short-circuited stubs are then transformed to the middle of the equivalent circuit by applying Kuroda s identity of the second kind [13], as shown in Fig. 4(b). It is now clear that, in order to maintain balanced outputs from port 2 and port 3 for a broad bandwidth, the unit elements attached to these two (1) (2) (3) The three-coupled lines need not be symmetrical. To synthesize the balun, the equivalent circuit shown in Fig. 4(a) will be further simplified. Under the conditions of (4) and (5), the middle of the short-circuited stub with characteristic admittance of between port 2 and port 3 can be treated as a virtual ground. Therefore, it can be separated into two equal parts and then be absorbed by the short-circuited stubs connected to port 2 and port 3. Furthermore, the short-circuited stub attached to port 1 in Fig. 4(a) is also divided into two parts, and, so that the input admittances looking at port 1 to the left-hand side and to the right-hand side would be the same. With the help of equivalent circuit shown in Fig. 4(b), the admittances and can be obtained by solving (8), shown at the bottom of this page, and the results are (9) (10) as shown in Fig. 4(c). Now the circuits on the two sides are identical two-port networks. Therefore, the synthesis of a three-port balun circuit is reduced to the synthesis of a two-port network, which will be discussed in Section III, as shown in Fig. 4(d). It should be noted that in this two-port network model, the load connected to port 2 should be twice the admittance that is connected to port 2. Follow the above procedure and use the network transformation in [11], the sufficient conditions for balanced outputs are found as (11) (12) (8)

3 142 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 Fig. 4. (a) (c) Equivalent circuits of type-i balun. (d) Network model for balun synthesis. Fig. 5. Equivalent network models of: (a) type-ii and (b) type-iii baluns. for the type-ii balun, and (13) for the type-iii balun, respectively. Equations (11) and (12) are equivalent to (14) (15) which means that the capacitance between line 1 and ground should be equal to that between line 3 and ground, and line 2 needs to be electrically shielded from ground, just as type 3 and type 10a baluns in [9]. The three-coupled lines are also not required to be symmetrical. For the type-iii balun, the condition of (13) can be satisfied by and. It then becomes a type 4c balun in [9], and its planar circuit implementation in [10]. They have line 3 electrically shielded from the other two lines by ground and line 1 electrically shielded from ground by line 2. It should be noted that there are still many other possibilities to satisfy (13). To design these two types of three-line baluns, one may follow the same method outlined above and find the final equivalent circuits as shown in Fig. 5. They have the same network configurations as that in Fig. 4(d). Therefore, the syntheses of these three types of three-line baluns are the same. The network model in Fig. 5(b) also shows that the Fig. 6. Equivalent circuits for the three-balun synthesis with: (a) second-, (b) third- (case a), (c) third- (case b), and (d) fourth-order insertion loss functions. three-coupled lines of the type-iii balun should not be symmetrical because the admittance of the unit element will be zero. This will result in an all-stop response. III. SYNTHESIS PROCEDURE OF THE THREE-LINE BALUNS The synthesis procedure of the three-line baluns is similar to that of the broadband matching design for a load composed of a conductance shunted by a short-circuited stub [14]. After applying the Kuroda s identity of the second kind, the left-hand-side short-circuited stub of the network model shown in Fig. 4(d) can be transformed to the right-hand side, and the result is correspondent with the matching network, which has a second-order insertion loss function. Furthermore, with the additional input and output transmission-line sections, the baluns with third- and fourth-order insertion loss functions can also be obtained, as shown in Fig. 6, where the corresponding characteristic admittances,, and for type-i to type-iii

4 LEE AND TSAI: EXACT SYNTHESIS OF BROADBAND THREE-LINE BALUNS 143 TABLE I REPRESENTATIONS OF Y, Y, AND Y FOR TYPE-I TYPE-III BALUNS Fig. 7. Equivalent circuit of the balun designs with alternative terminations. baluns are given in Table I. It should be noted that the output resistance in this equivalent circuit is one fourth of the resistance at the balanced output of the balun. All the transmission-line sections must also be commensurate, i.e., at the design frequency. The general form of th-order Chebyshev insertion loss function for the networks shown in Fig. 6 is given by where is the selected zero and is the left-hand-side pole in conjugate pairs for, and is a constant to ensure. Generally, there are two choices of for and four for and with a given Chebyshev insertion loss function. For the case of having perfect matches in the passband, i.e.,, all zeros will lie on the axis, thus, there are no alternative choices. Therefore, the expressions of for and in [14] are not complete since Hurwitz polynomials are assumed. The output reflection coefficient can also be found as (19) The input and output admittances can be calculated by (16) where is the Chebyshev polynomial of the first kind of degree, and and are the electrical length of the transmission lines and that at the cutoff frequency, respectively. The minimum and maximum of in the passband are and, respectively. By applying the Richards transformation to (16), the squared modulus reflection coefficient can be expressed by (20) (21) The characteristic admittance of each transmission-line section of the equivalent circuit can then be evaluated with the derived and. The details are discussed as follows. A. The input admittance of the circuit shown in Fig. 6(a) in terms of the characteristic admittances of the transmission-line sections can be expressed by (17) where the coefficient and for and can be found in [14], and those for are given in this paper. They are all summarized in the Appendix. The zeros and poles of (17) need to be solved to determine its constituents and. In order to have realizable networks, the denominator of,, should be a strictly Hurwitz polynomial, i.e., all the left-hand-side poles in the -plane are chosen to determine. However, the numerator is not necessary to be Hurwitz, and the zeros can be selected in conjugate pairs in either the left- or right-hand side of the plane. Therefore, the input reflection coefficient can be written in the form of (18) (22) Comparison between the corresponding coefficients gives the characteristic admittances as (23) (24) (25) B. (case a) For the third-order balun design with a transmission line at the input (case a), as shown in Fig. 6(b), the input unit

5 144 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 TABLE II CIRCUIT PARAMETERS OF THE BALUN DESIGNS FOR n =2TO n =4WITH js j = 020 db AND js j = 025 db IN THE PASSBAND element can be extracted by applying Richards theorem to, i.e., (26) The input admittance of the remaining network is then solved by removing the unit element as (27) Although the numerator and denominator of appear to be one degree higher than those of, they have a common factor. Therefore, is reduced to the formation given in (22). The characteristic admittances,, and are then obtained accordingly. C. (case b) The equivalent circuit of the third-order balun design in case b shown in Fig. 6(c) is similar to that of case a, whereas the unit elements and short-circuited stubs are presented in reverse order. Therefore, the output transmission line can be derived by (28) The output admittance of the rest of the circuit are then calculated by which can be used to obtain,, and. (29) D. The balun design with fourth-order response is shown in Fig. 6(d), where additional unit elements are connected to both the input and output of the three-line balun. Similarly, applying (26) and (28) gives the characteristic admittances of these two unit elements. After removing the input unit element using (27), the input admittance of the rest of the circuit is obtained

6 LEE AND TSAI: EXACT SYNTHESIS OF BROADBAND THREE-LINE BALUNS 145 TABLE III CIRCUIT PARAMETERS OF THE BALUN DESIGNS FOR n =2TO n =4WITH js j = 020 db AND js j = 01 db IN THE PASSBAND and can be expressed by (30), shown at the bottom of this page. Comparing the coefficients in (30) gives the solutions for the characteristic admittances of the rest of the transmission-line sections as follows: (31) (32) are 2 and 0.5, respectively. The new parameters are calculated as shown in this figure, and the results are,,,, and. The characteristic admittances,, and are now known and they must be realized by three-coupled lines. In order to obtain a realizable circuit, the characteristic admittances for type-i balun should follow the following criteria: (33) For convenience, the circuit parameters of balun designs for to with and are calculated and summarized in Tables II and III. The baluns in Table II have db and db in the passband, and those in Table III have db and db. For the balun designs with other values of and, the parameters can also be derived from these tables. By adding redundancy unit elements to the input and output, the circuits in Fig. 6(a) (c) can all be put into the same form, as shown in Fig. 6(d). It is then transformed to its equivalent circuit as shown in Fig. 7 by using Kuroda s identity of the second kind. Redundancy unit elements at the two ends, if they exist, can now be removed. This completes the impedance transformation. For example, a fourth-order balun with and is designed to have a fractional bandwidth of 120% and db and db in the passband, the admittances would be,,,, and.ifa balun design with and is needed, the corresponding load impedance ratios and in Fig. 7 The criteria can also be found as for the type-ii balun and (34) (35) (36) for the type-iii balun. For example, is found for the above balun design with and and leads to unrealizable three-coupled lines for type-ii and type-iii baluns. Moreover, type-i and type-ii baluns are not applicable (30)

7 146 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 to the balun design with and since is obtained. Therefore, the derived admittances in Tables II and III should be carefully checked with the criteria (34) (36) for realization by different types of three-line baluns. The admittances,, and represent different characteristics of the three-coupled lines of these three types of baluns. As examples, in type-i and type-ii baluns can be found as and (37) (38) respectively, and is not restricted to be positive. The condition of for the type-i balun usually means that the balun has a high impedance of line 2 or tight coupling between line 1 and line 3. To ease the implementation, a unit element with characteristic admittance of added at the input is helpful. It is observed that adding the unit element with characteristic admittance of at the output would increase, which represents the coupling between line 1 and line 2, and line 1 and line 3 for type-i balun and the admittance of for type-ii balun. The characteristic admittance shows the coupling between line 2 and line 3, and line 1 and line 3 for the type-i balun, the coupling between line 2 and line 3 for the type-ii balun, and the difference between the coupling of line 1 and 2 and line 2 and 3 for the type-iii balun, and will also be increased due to the presence of the unit element at either the input or the output. Baluns with the same amplitude response could have more than one realization since the selections of zeros of are not unique. It is found that the determination of with all the left-hand-side zeros in the -plane will lead to the highest and lowest, and the results are contrary if all the right-hand-side zeros are selected. IV. DESIGN EXAMPLE A third-order balun (case a) with 50:100- terminations from the unbalanced to the balanced port is presented as an example, and the corresponding terminations in the equivalent circuit shown in Fig. 6(b) are and. The passband central frequency is 2 GHz with a fractional bandwidth of 80%, which yields, and the maximum and minimum of in the passband are 20 db and 25 db, respectively. and are then calculated to be and , respectively. can now be evaluated using (A3) (A6) and is given by Fig. 8. (a) Circuit parameters and layout of the balun design example. (b) Dimensions of the three-line balun in mil: W =90:6, W = 222, W =39:4, S =12:6, S =96:1, L = L = L = 866. All the zeros and poles in the left-hand side of the -plane are chosen for in this example. After applying (18) and (20), the input admittance becomes (42) The characteristic admittance of the input unit element is calculated by employing Richards theorem as (26) (43) The input admittance of the network after the removal of the input unit element can be derived by (27), and the result is (44) Solving (39) gives the zeros of and poles of (39) (40) (41) The common factor in (44) is then cancelled, and the characteristic admittances of the remaining transmission-line sections are obtained by (23) (25) as follows: (45) These results can, of course, be read out directly from Table II. Apparently, the condition of is obtained in this design and its implementation using type-ii and type-iii baluns is not possible. Therefore, this design is realized by the type-i

8 LEE AND TSAI: EXACT SYNTHESIS OF BROADBAND THREE-LINE BALUNS 147 Fig. 10. Sensitivity analysis of the balun design. (a) Magnitude balance. (b) Phase balance. Fig. 9. (a) Simulated and (b) measured response of the balun design. (c) Differences of the transmission magnitude and phase. the electromagnetic simulation results are compared to the ideal response and it shows only a minor deviation in the bandwidth. The measured results shown in Fig. 9(b) have approximately 3.6-dB transmission loss around the central frequency. The simulated and measured imbalance of the magnitude and phase is given in Fig. 9(c), where the measured magnitude imbalance is within 0.6 db in the passband, and the phase error is less than 4. The results of sensitivity analysis are given in Fig. 10. The width of line 3, i.e.,, has more effects on the magnitude and phase balance. Still, the measured results agree well with the simulated response, and there is only a little discrepancy between them at the higher passband. balun. Actually, for the same specification of bandwidth and return loss, type-ii and type-iii baluns must be implemented with (case b) and. Moreover, the spacing between the three-coupled lines of type-ii and type-iii baluns would be much smaller than that of the type-i balun, and this makes their realization more difficult. The balun is implemented using a multilayer Rogers RO3003 substrate with a total thickness of 90 mil, a dielectric constant of 3, and a loss tangent of The circuit layout and dimensions of the three-line balun are shown in Fig. 8, where line 1 and line 2 of the balun are fabricated on the middle layer, and line 3 is on the upper layer. The thickness between these two layers is 10 mil. Fig. 9(a) and (b) show the simulated and measured reflection and transmission coefficients, respectively. In Fig. 9(a), V. CONCLUSION The design procedure of three types of three-line baluns has been studied in this paper. The equivalent circuits are derived by using the model of coupled transmission lines. It is found that the equivalent circuits of these three types of baluns have the same network configuration. Therefore, the syntheses for these three types of baluns are the same and their design methods are similar. The synthesis procedure has been derived for the baluns with second- to fourth-order Chebyshev response. Some design guidelines are also presented. A third-order balun design example is used to demonstrate the proposed method, and the measurement results are in good agreement with the theoretical study.

9 148 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 57, NO. 1, JANUARY 2009 APPENDIX By observing (16) and (17), the expressions of the numerator and denominator of are almost the same, except the constants and. Therefore, only the numerators of for to are given. The denominators can be derived by replacing in the numerators by. : The numerator of (17) is where where : The numerator of (17) for is given by (A1) (A2) (A3) (A4) (A5) (A6) [2] J. H. Cloete, Exact design of the Marchand balun, Microw. J., vol. 23, no. 5, pp , May [3] J. H. Cloete, Graphs of circuit elements for the Marchand balun, Microw. J., vol. 24, no. 5, pp , May [4] A. M. Pavio and A. Kikel, A monolithic or hybrid broadband compensated balun, in IEEE MTT-S Int. Microw. Symp. Dig., 1990, vol. 1, pp [5] K. Nishikawa, I. Toyoda, and T. Tokumitsu, Compact and broad-band three-dimensional MMIC balun, IEEE Trans. Microw. Theory Tech., vol. 47, no. 1, pp , Jan [6] W. M. Fathelbab and M. B. Steer, New classes of miniaturized planar Marchand baluns, IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp , Apr [7] C. M. Tsai and K. C. Gupta, CAD procedures for planar re-entrant type couplers and three-line baluns, in IEEE MTT-S Int. Microw. Symp. Dig., 1993, vol. 2, pp [8] C. Cho and K. C. Gupta, A new design procedure for single-layer and two-layer three-line baluns, IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp , Dec [9] R. C. Johnson and H. Jasik, Antenna Engineering Handbook, 2nd ed. New York: McGraw-Hill, 1984, p [10] B. Lee, D. Park, S. Park, and M. park, Design of new three-line balun and its implementation using multilayer configuration, IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp , Apr [11] R. Sato and E. G. Cristal, Simplified analysis of coupled transmissionline networks, IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 3, pp , Mar [12] D. W. Kammler, Calculation of characteristic admittances and coupling coefficients for strip transmission lines, IEEE Trans. Microw. Theory Tech., vol. MTT-16, no. 11, pp , Nov [13] P. A. Rizzi, Microwave Engineering, Passive Circuits. Englewood Cliffs, NJ: Prentice-Hall, 1988, pp [14] R. Levy and J. Helszajn, Specific equations for one and two section quarter-wave matching networks for stub-resistor loads, IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 1, pp , Jan where : For, the numerator of (17) is (A7) (A8) (A9) Hong-Ming Lee (S 03 M 06) was born in Nantou, Taiwan. He received the B.S. and Ph.D. degrees in electrical engineering from National Cheng Kung University, Tainan, Taiwan, in 2002 and 2006, respectively. He is currently a Post-Doctoral Researcher with the Institute of Computer and Communication Engineering, Department of Electrical Engineering, National Cheng Kung University. His research interests include microwave passive components and measurements. (A10) (A11) REFERENCES [1] N. Marchand, Transmission-line conversion transformers, Electronics, vol. 17, pp , Dec Chih-Ming Tsai (S 92 M 94) received the B.S. degree in electrical engineering from National Tsing Hua University, Taiwan, in 1987, the M.S. degree in electrical engineering from the Polytechnic University, Brooklyn, NY, in 1991, and the Ph.D. degree in electrical engineering from the University of Colorado at Boulder, in From 1987 to 1989, he was a Member of the Technical Staff with Microelectronic Technology Inc., Taiwan, where he was involved with the design of digital microwave radios. In 1994, he joined the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, where he is currently a Professor. His research interests include microwave passive components, high-speed digital design, and measurements.